residue theorem

The residue theorem is a powerful tool from complex analysis that helps evaluate certain types of integrals involving complex functions. When you integrate a complex function around a closed loop (path), the theorem states that the integral depends only on the singularities (points where the function behaves badly) inside the loop. Specifically, it calculates the total contribution (residue) of these singularities, simplifying the integral to a sum of these residues multiplied by 2πi. This approach transforms complex contour integrals into manageable algebraic calculations, making it easier to evaluate integrals that are difficult to handle directly.